particle properties

space time: angular momentum, Positions


Particles have properties that seemed tied to some internal structure. This is similar to the color of a ball. The property is an inherent feature of the ball.Its motion (momentum) is tied to its space-time properties.Internal properties are independent of space time


Intrinsic properties standard

charge, mass(Usually we choose eigenstates of definite mass and charge) [Mass referred to here is the rest mass of the particle.]


Spin seems to bridge the gap. There is some internal structure associated with particles that provides the particle with an intrinsic angular momentum.This behaves in the standard way that angular momentum behaves and must be added to orbital angular momentum. An electron can have its spin reoriented from right to left pointing.This constitutes a change of angular momentum of 1 unit [].If this were part of the change associated with an atomic transition then the electromagnetic field would need to have the compensating angular momentum change to keep angular momentum conserved.


Rotations are our go to transformations:


There are unitary operators that rotate quantum states.

There are unitary operators that translate states.


Unitary is the complex space analog of orthogonal. Rotations do not change the length of a vector. Unitary transformations do not change the ďamountĒ of quantum states. This can be more formally stated in terms of probability.


Note: If you choose the standard position space representation for the momentum operator where for simplicity we consider only one dimension and if we remember the Taylor expansion as the tool to interpret. One can show that the translation operator will move a function a distance.


Quantum numbers

momentum, angular momentum, spin, mass, charge, flavorŤ(isospin, strangness, charm), electron number, parity, energy, lepton number, baryon number,


File:Elementary particle interactions.svg



  • matrices are operators and that indeed they operate on a column vector to produce a different column vector.

matrix operates to produce new column

matrix operates but the result is just a constant times the initial columnŤ eigenvector of the operator.

operator has no inverse. There are various properties that can be ascertained as to the behavior of operators.


  • derivatives are operators and they operate on functions to produce new functions

derivative operates to produce new function

derivative operates but the result is just a constant times the initial functionŤ eigenvector of the operator.



†† form of the rotation operator where J is the angular momentum operator.

Operators donít commute so we label are states with the values of total and z-component.

Raising and lowering or ladder operators.

These operators produce a new state that has the z-component of angular momentum increased/decreased by 1.


The key details concerning structure is that are a subset of operators and associated quantum numbers to label multiplets and there are operators that move you through the multiplet.


We can build new multiplets by combining multiplets. Two spin particles can be combined to for a composite system (hydrogen atom)

use individual quantum numbers to label state

use the total angular momentum and the total z-projection.

Both sets of states span the space.

individual Q# for spin 1/2

total Q# for spin 1/2


Review: [various web sources]

Particles -------------------------------



spin 1/2







spin 1 (graviton spin 2)

interactions vector bosons




Gravity†††††††††† G


Higgs Boson

Standard model of elementary particles. The electron is at lower left.







File:Baryon decuplet.svg